2
Study of Inductive-Capacitive Series Circuits
Using the Simulink Software Package
Titu Niculescu
University of Petrosani
Romania
1. Introduction
This chapter presents a modern method for approaching the electrical circuits using the
MATLAB-SIMULINK package programs. The simple series circuits which are switched on a
DC voltage at the initial moment are presented below. For the young researchers these are
very useful. We can determine the current variation forms and the reactive elements
voltage, by using this virtual medium. Each presented case contains an analytical
presentation of the problem, but it also contains electrical diagrams of electrical parameters.
The diagrams were obtained by different methods which use this programs package.
2. Study of the inductive series circuits
We will consider the RL series circuit with concentrated parameters from Fig.1. At the initial
moment, the k circuit switcher is closed and we intend to study the behavior and variation
of circuit electrical parameters after connection.
Fig. 1. RL series circuits at closing
2.1 Theoretical study of the circuit
The differential equation which corresponds to the transitory regime immediately after
closing is the following:
Ri  L
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di
E
dt
(1)
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Technology and Engineering Applications of Simulink
The current expression through the circuit after closing is the solution of the differential
equation of the circuit (1).
i( t ) 

E
(1 - e T )
R
t
(2)
where:
T
L
R
(3)
and it represents the time constant of the circuit.
The voltage expression on the coil after connection is:
u(t )  L

di
 Ee T
dt
t
(4)
2.2 SIMULINK model of the circuit
The SIMULINK model of the circuit after closing was done on grounds of equation (1)
where the derivative of the current was separated.
di 1
 (E - Ri )
dt L
(5)
The SIMULINK model obtained in this way is shown in Fig.2 and was created in order to
allow the drawing of the current diagrams through the circuit and the voltage on the coil,
for different values of the R resistance and the E D.C. voltage.
1
Step
Add
1/L
1
s
Integrator
Scope I
du/dt
Product
Derivative
50
Scope U
1
L
R
Fig. 2. SIMULINK model of the RL circuit at closing
The k switcher is realized using a voltage step signal which is applied at the terminal of the
circuit, to simulate the closing of the k switcher.
If we consider the following values of the electric parameters
-
D.C. voltage applied: E = 24[V];
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Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package
-
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The resistance value: R = 50[Ω];
The circuit inductivity: L = 1[H],
we will obtain the diagrams from Fig.3 and Fig.4.
Fig. 3. Variation of the coil voltage after closing
Fig. 4. Variation of the circuit current
2.3 Analysis of RL series circuits using the SimPowerSistems software package
The SimPowerSystems software package is a component of the MATLAB program which
allows analyzing an electrical circuit by drawing it in an editing window. The
SimPowerSystems model is presented in Fig.5 and allows the visualization of the circuit
current and the coil voltage if we apply a step voltage signal and a unitary impulse. The
electrical parameters of the circuit have the values:
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Technology and Engineering Applications of Simulink
R = 50[Ω]; L = 1[H].
C ontrolled Voltage Source
R
L
C urrent Measurement
s
-
+
+
Scope I
i
-
Step
+
v
C o ntinuo us
Voltage Measurement
Scope U
po we rgui
Fig. 5. SimPowerSystems model of an inductive circuit
The Continous powergui analyze block is required to be present in the drawing window for
launching the simulation model. We will open an analysis window by double-clicking on
this block, Fig. 6.
Fig. 6. The analysis window of the circuit
The selection of the Use LTI Viewer button will lead to the opening of the window in Fig.7,
from which we can select the current diagram or the coil voltage diagram.
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Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package
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Fig. 7. The LTI Viewer window
The selection of U_Voltage Measurement or I_Current Measurement is followed by the
obtaining of the coil voltage diagram and of the circuit current upon applying the step
signal and a unitary impulse. These are presented in Fig.8 and Fig.9.
Fig. 8. The coil voltage variation upon applying a step signal and a unitary impulse.
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Technology and Engineering Applications of Simulink
Fig. 9. The current variation upon applying a step signal and a unitary impulse.
We can use the Edit/Plot Configurations and Edit/ Viewer Preferences options, in order to create
these diagrams. We can also select the Bode option from these menus, which allows the
obtaining of the current frequency diagrams of the RL series circuit.
Fig. 10. The amplitude frequency diagram of the voltage coil.
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Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package
25
Fig. 11. The frequency current diagrams.
3. Study of the capacitive series circuits
We will consider the RC series circuit with concentrated parameters from Fig.12. At the
initial moment, the k circuit switcher is closed and we want to study the current variation
through the circuit and the capacitor voltage variation, after connection.
Fig. 12. RC series circuit at closing
3.1 Theoretical study of the circuit
The integral equation which corresponds to the transitory regime immediately after closing
is given by:
Ri 
1
C

idt  E
Equation (6) is equivalent with differential equation:
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(6)
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Technology and Engineering Applications of Simulink
RC
du
uE
dt
(7)
The expression of the capacitor voltage after closing is the solution of differential equation
(7):
u(t )  E(1 - e

t
T
)
(8)
du E  T
 e
dt R
(9)
where T  RC , represents the time constant of the circuit.
The circuit current after closing is given by:
i C
t
3.2 SIMULINK model of the circuit
The SIMULINK model of the circuit after closing was done on grounds of equation (7)
where the voltage derivative was separated.
du
1

(E - u)
dt RC
(10)
The SIMULINK model obtained is shown in Fig.13 and was created in order to allow the
drawing of the current diagrams through the circuit and the voltage capacitor, for different
values of the R resistance, of the E D.C. voltage and of the C capacitor.
Add
1
s
Step
Scope U
Integrator
Product1
du/dt
Derivative
1500
Product2
Scope I
R
-CC
Product
Divide
1
1
Fig. 13. SIMULINK model of the RC circuit at closing
The k switcher is realized using the same voltage step signal which is applied at the terminal
of the circuit, to simulate the closing of the k switcher.
If we consider the following values of the electric parameters:
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27
- D.C. voltage applied: E = 24[V];
- The circuit resistance: R = 50[Ω];
- The circuit capacity: C = 100[µF],
we will obtain the diagrams from Fig.14 and Fig.15.
Fig. 14. The capacitor voltage variation
Fig. 15. The current variation after closing
3.3 Analysis of RC series circuits using the SimPowerSistems software package
The SimPowerSystems model is presented in Fig.16 and allows the visualization of the
circuit current and the capacitor voltage if we apply a step voltage signal and a unitary
impulse.
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Technology and Engineering Applications of Simulink
C ontrolled Voltage Source
R
C
C urrent Measurement
s
Scope I
+ i
-
+
Step
+
- v
Voltage Measurement
Scope U
C o ntinuous
powe rgui
Fig. 16. SimPowerSystems model of the capacitive circuit
The electrical parameters of the circuit have the values:
-
R = 50[Ω]; C = 10[µF].
To study the voltage variation on the capacitor, we follow the steps:



Select by double clicking the Continous powerguy block from the simulation model.
Select Use LTI Wiewer in the new window;
We will select U Voltage Measurement.
This method allows us to obtain the voltage capacitor diagram upon applying the step
signal and the unitary impulse (Fig.17).
The current variation through the circuit upon applying a step signal and a unitary impulse
was obtained by selecting I_Current Measurement option (Fig.18).
Fig. 17. The capacitor voltage variation upon applying a step signal and a unitary impulse.
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Fig. 18. The current variation upon applying a step signal and a unitary impulse.
The Bode diagrams which allow the visualization of the frequency behavior of the circuit are
presented bellow:
Fig. 19. The capacitor voltage frequency diagrams
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Technology and Engineering Applications of Simulink
Fig. 20. The current frequency diagrams.
4. Study of the inductive-capacitive series circuits
We will consider the RLC series circuit with concentrated parameters from Fig.21. At the
initial moment, the k circuit switcher is closed and we intend to study the behavior and
variation of circuit electrical parameters after connection.
Fig. 21. RLC series circuit at connecting
4.1 Theoretical study of the circuit
The integral-differential equation which corresponds to the transitory regime of the
considered circuit is the following:
Ri  L
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di 1
idt  E

dt C 
(11)
Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package
31
or:
LC
d 2u
du
 RC
u E
dt
dt 2
(12)
The following notations are made:

R
2L
(13)
The circuit amortization:
0 
1
LC
(14)
the circuit personal pulsation:
   2 - 02
(15)
We will consider the situation when   0 or R  2 L C , which will be checked by the
resistor in the circuit.
Consequently, the solving of the differential equation gives the following solutions:


1
u(t )  E [1 (r1 er2 t - r2 e r1t )]


2  2 - 02


i( t ) 
E
2 L  2 - 02
e
r1 t
- er2 t

(16)
(17)
where r1 and r2 are the roots of the characteristic equation:
r1  -  
r2  - - 
(18)
We will consider the situation when δ <ω0 or R  2 L C . The following notation was made:
02 -  2   '2
(19)
Consequently, the solving of the differential equation gives the following solutions:



u(t )  E [1 - 0  e- t  sin( ' t   ')]
'


i( t ) 
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E - t
e sin  ' t
 'L
(20)
(21)
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Technology and Engineering Applications of Simulink
where
 '  arccos

0
(22)
4.2 SIMULINK model of the circuit
The SIMULINK model of the circuit after closing was done on grounds of the second order
differential equation (12), which is put in the form (23) where the higher order derivative is
separated:
d 2u
1 
du


[E - RC
- u]
dt
LC 
dt

(23)
The SIMULINK model from Fig.22 generates the voltage capacitor and the current through
the circuit during the transitory regime. To this purpose, two values of the R resistor are
considered, which correspond to two important regimes:


Aperiodic regime;
Oscillatory regime.
For each regime the capacitor voltage and the circuit current variation diagrams are
plotted.
-K-
Step
1/LC
1
s
1
s
Integrator
Integrator1
ScopeU
Add
-C1500
C1
R
RCdu/dt
Fig. 22. SIMULINK model of the circuit
If we consider:
-
The value of the D.C. voltage: E = 24 [V]
The circuit resistance: R = 1.5 [kΩ],
The coil inductance: L = 10-4 [H]
The capacitor capacity: C = 10 [nF],
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-KC
du/dt
Derivative
ScopeI
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we get the following MATLAB diagrams (Fig.23 and Fig.24) according to the equations
(16) and (17) for a-periodic mode, and according to the equations (20) and (21) for
oscillating mode (Fig.25 and Fig.26).
Fig. 23. The voltage capacitor variation in the a-periodic mode
Fig. 24. The current variation in the a-periodic mode
Fig. 25. The capacitor voltage variation in the oscillating regime
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Technology and Engineering Applications of Simulink
Fig. 26. The current variation in the oscillating regime
The k switcher is realized by using a voltage step signal which is applied at the terminal of
the circuit, to simulate the closing of the k switcher. Changing the value of resistance,
automatically leads to the updating of the voltage and current charts.
4.3. Analysis of inductive-capacitive circuits using the SimPowerSystems software
package
The SimPowerSystems software package allows the study of the circuit behavior, when it is
connected to a voltage step or a unitary impulse, using Continous powerguy block. The
current variation in the circuit, the voltage variation of the capacitive element, or the
frequency behavior of the circuit for different values of circuit elements can be studied.
4.3.1 SimPowerSystems model of the circuit
The MATLAB software package contains SimPowerSystems of Simulink, which can
simulate the electrical circuits and analyze different operating regimes. Because the signal
step response or pulse generates a transitory regime, the circuit behavior in these conditions
can be studied. The simulation is shown in Fig.27:
The advantage of this simulation method is that the circuit can be tested at both signals, unit
step and unitary impulse, and leads to several types of typical circuit diagrams. Two sets of
values were chosen in this case for circuit parameters, which correspond to two special
regimes.
In the simulation model from Fig.27 we used a Controlled Voltage Source controlled by a step
voltage signal. In this way, the switcher closing can be simulated.
The SimPowerSystems software package allows the obtaining of a lot of diagrams for a
circuit. So, we will study the capacitor voltage diagrams and current diagrams for the two
occurring regimes:


A-periodic regime:
Oscillating regime.
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Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package
R
L
s
+
+
Step
Controlled Voltage Source
Continuous
i
-
Current Measurement
ScopeI
Voltage Measurement
C
pow ergui
v
+
-
ScopeU
Fig. 27. SimPowerSystems model of the circuit
1.
Voltage capacitor
The a-periodic loading of the capacitor leads to the obtaining of the capacitor voltage
variation diagrams when the unit step and the unitary impulse are applied to the input.
These charts are presented below.
The following values of electrical parameters were considered:


-
For a-periodic regime:
E = 24 [V], the step voltage applied;
R = 1,5 [kΩ], the electrical circuit resistance;
L  10  4 [ H ] , the circuit inductance;
C = 10 [nF], the circuit capacity.
For oscillating regime:
E = 24 [V], the step voltage applied;
R = 10 [Ω], the electrical circuit resistance;
L  10  4 [ H ] , the circuit inductance;
C = 10 [nF], the circuit capacity.
To study the capacitor voltage variation, we follow the steps:
-
Select by double clicking the Continous powerguy block from the simulation model;
Select Use LTI Wiewer will appear in the new window;
We will select U Voltage Measurement.
The POWERGUY analysis block, with the above values, allows the obtaining of the
following Matlab diagrams (Fig.28 and Fig.29):
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Technology and Engineering Applications of Simulink
Fig. 28. The capacitor voltage variation in the a-periodic mode
Fig. 29. The capacitor voltage variation in the oscillating regime
The POWERGUY block permits the analysis and plotting of the frequency characteristics of
the capacitor voltage, where the frequency in logarithmic coordinates was considered on the
horizontal axis (BODE diagram).
For this, we follow these steps:
-
Select by double clicking Continous powerguy block ;
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Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package
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Use LTI Wiewer will appear in the window and we will select it;
In the occurring window we will select U voltage Measurement;
Now, we will select Edit / Plot Configuration / Bode
Fig. 30. Frequency diagrams of the voltage swing on the capacitive element
2.
Study of circuit current
To study the circuit current variation the same procedure is used, mentioning that instead of
selecting the U voltage Measurement, I Current Measurement must be selected.
A-periodic loading of the capacitor leads to obtaining the current variation diagrams when
the unit step and the unitary impulse are applied to the input.
For this, we will follow these steps:
-
Select Continous powerguy block diagram simulation by double click;
Select Use LTI Wiewer in the appeared window;
Select I Current Measurement in the window appeared;
Select Edit / Plot Configuration / Bode
The SimPowerSystems software package allows the obtaining of two diagrams for a circuit.
So, we will study the current variation diagrams for the two occurring regimes:


A-periodical regime:
Oscillating regime.
The a-periodical and oscillating regime according to these values are obtained in the
following charts (Fig.31 and Fig.32):
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Technology and Engineering Applications of Simulink
Fig. 31. The current variation for signal step and impulse in a-periodic mode
Fig. 32. The current variation for signal step and impulse in the oscillating regime
The POWERGUY block allows the analysis and plotting of frequency current characteristics,
where the frequency in logarithmic coordinates was considered on the horizontal axis
(BODE diagram). For this, we will follow these steps:
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Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package
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Select Continous powerguy by double clicking the block diagram simulation;
Select Use LTI Wiewer in the appeared window;
Select I Current Measurement in the window appeared;
Select Edit / Plot Configuration / Bode
Fig. 33. Frequency diagrams of the current swing through the circuit
In the diagram from Fig.33 we can observe a maximum of current which correspond to a
resonance regime. We can determine the resonance frequencies for an inductive-capacitive
circuit using the simulation circuit model and the SimPowerSystems software package. With
POWERGUY block, other electrical circuit parameters can be determined.
5. Conclusions
An important conclusion of this chapter is that the electrical circuits, regardless of
configuration, can be studied with this modern method which involves the use of virtual
medium.
Three elementary circuits are presented here, but the study modality can be extended to
other circuits configuration as well.
If we consider an electric circuit, a first problem is that of writing the characteristic integral
differential equations, which can be integrated with a simulation model and which uses the
Simulink software package. These models can be conceived so that the electric circuit
parameters must be electric input values in the simulation model. Thus, the obtained
diagrams are automatically updated to any changes of circuit electrical parameters, and the
influence of each parameter variation in the final diagrams can be studied.
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Technology and Engineering Applications of Simulink
In order to realize the simulating model which integrates the differential equations, the basic
idea is to separate in the left member of equation, the higher order derivative. The analytical
expressions obtained in the right member of the equation underlie the achievement of the
simulating model by means of specific blocks of the virtual medium. In the case of complex
circuits, the simulating model is conceived on a differential equations system.
Another given facility of this package programs, is the possibility to study the circuit
behavior upon applying the standard signals (step unit, unitary impulse). The specific
analysis block Continous Powerguy allows the circuit response to these signals. It also allows
the study of circuit frequency characteristics and the other specific diagrams: the coil
hysteresis diagrams, line parameters, FTF analysis, Bode Diagrams, Nyquist diagrams,
Nichols diagrams and others.
This study intends to be a starting point in the approach of more complex circuits.
6. References
Niculescu, T.; Pasculescu, D.; Pana, L. (2010). Study of the operating states of intrinsic safety
Barriers of the electric equipment intended for use in atmospheres with explosion
hazard,Main Page of the Journal WSEAS TRANSACTIONS on CIRCUITS and
SYSTEMS, ISSN: 1109-2734,
Pasculescu, D.; Niculescu, T.; Pana, L. (2010). Uses of Matlab software to size intrinsic safety
barriers of the electric equipment intended for use in atmospheres with explosion,
International Conference on ENERGY and ENVIRONME TECHNOLOGIES and
EQUIPMENT (EEETE '10), ISBN: 978-960-474-181-6, Bucharest, Romania
Niculescu, T.; Niculescu, M. (2010). The study of fundamental electrical circuits transitory
Phenomena
using
MATLAB
software,
UNIVERSITARIA
SIMPRO
ElectricalEngineering and Energetics System Control, Applied Informatics and
ComputerEngineering, ISSN 1842-4449 Petrosani, Romania
Ghinea, M.; Firţeanu, V. (1999). MATLAB calcul numeric, grafică, aplcatii, Teora, ISBN 973-601275-1, Bucharest, Romania
Halunga-Fratu, S.; Fratu, O. (2004) Simularea sistemelor de transmisiune analogice şi digitale
folosind mediul MATLAB/SIMULINK, MatrixRom, ISBN 973-685-716-6, Bucharest,
Romania
Tudorache, T. (2006). Medii de calcul in ingineria electrica MATLAB, MatrixRom, ISBN 973775-005-6, Bucharest, Romania
Niculescu, T.;Costinaş, S. (1998) Electrotehnică, Printech, ISBN 973-9402-14-3, Bucharest,
Romania
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Technology and Engineering Applications of Simulink
Edited by Prof. Subhas Chakravarty
ISBN 978-953-51-0635-7
Hard cover, 256 pages
Publisher InTech
Published online 23, May, 2012
Published in print edition May, 2012
Building on MATLAB (the language of technical computing), Simulink provides a platform for engineers to plan,
model, design, simulate, test and implement complex electromechanical, dynamic control, signal processing
and communication systems. Simulink-Matlab combination is very useful for developing algorithms, GUI
assisted creation of block diagrams and realisation of interactive simulation based designs. The eleven
chapters of the book demonstrate the power and capabilities of Simulink to solve engineering problems with
varied degree of complexity in the virtual environment.
How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following:
Titu Niculescu (2012). Study of Inductive-Capacitive Series Circuits Using the Simulink Software Package,
Technology and Engineering Applications of Simulink, Prof. Subhas Chakravarty (Ed.), ISBN: 978-953-510635-7, InTech, Available from: http://www.intechopen.com/books/technology-and-engineering-applications-ofsimulink/study-of-inductive-capacitive-series-circuits-using-the-simulink-software-package
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